Powers and roots — squares, cubes and higher

Powers (also called indices or exponents) are shorthand for repeated multiplication. Roots are the inverse: they "undo" a power. Memorising a small bank of common values is the fastest route to the marks.

Index notation

a^n means a multiplied by itself n times, where n is a positive integer. Here a is the base and n is the index (or power).

• 5² = 5 × 5 = 25 — read as "5 squared".
• 5³ = 5 × 5 × 5 = 125 — "5 cubed".
• 2⁴ = 2 × 2 × 2 × 2 = 16 — "2 to the fourth".

Two important conventions:

• a¹ = a (any number to the first power is itself).
• a⁰ = 1 for any non-zero a. (We'll see why in N7.)

Square numbers and square roots

The squares you should know by heart: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 (1² to 15²).

The square root of x is the number that, when squared, gives x.

• √81 = 9 (because 9² = 81).
• √144 = 12.

⚠ Equation vs symbol: when solving x² = 49, write x = ±7. The symbol √49 alone refers to the positive root, 7.

Cube numbers and cube roots

Cubes 1³ to 5³ (and a few more): 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

³√125 = 5 (because 5³ = 125). Cube roots have only one real value, so the sign is preserved: ³√(−27) = −3.

Higher powers

• 2⁵ = 32, 2⁶ = 64, 2⁷ = 128, 2⁸ = 256, 2¹⁰ = 1024.
• 3⁴ = 81, 5⁴ = 625, 10⁴ = 10,000, 10⁶ = 1,000,000.

Sign rules for powers

• A positive base raised to any power is positive.
• A negative base raised to an EVEN power is positive: (−3)² = 9, (−2)⁴ = 16.
• A negative base raised to an ODD power stays negative: (−3)³ = −27, (−2)⁵ = −32.

⚠ Watch the brackets: −3² = −9 (only the 3 is squared) but (−3)² = 9.

Order of operations (BIDMAS)

Powers/Indices come before multiplication and division. So 2 × 5² = 2 × 25 = 50, not 100. Roots also count as indices for BIDMAS.

Estimating roots that aren't whole

Many square roots aren't whole numbers, but you can estimate:

• √50: between √49 = 7 and √64 = 8, closer to 7. So roughly 7.07.
• √120: between 10 and 11. Try 10.95² ≈ 119.9 — about right.